New Gauge Symmetries from String Theory
Pei-Ming Ho
Physics Department
National Taiwan University
Sep. 23, 2011@CQSE
Gauge vs Global
For covariant quantities
Old definitions in some textbooks:
U = independent of spacetime global U = function of spacetime gauge (local)
Gauge vs Global (2)
• Translation symmetry is usually considered as a global symmetry.
• Different interpretation of the transformation:
Translation by a specific length L can be
“gauged” (defined as a gauge symmetry)
equivalence:
(compactification on a circle).
• Gauge potential is useful but not necessary.
x(t) » x(t) + L
Gauge vs Global (3)
• 2nd example:
base space = R+ with Neumann B.C. at x = 0.
• 3rd example:
space an interval of length L/2 w.
Neumann B.C.
Space/(subgroups of isometry) orbifolds
x® - x, f(x) ® f(-x)
Gauge vs Global (4)
New (better) definition:
• Gauge Symmetry
– Transformation does not change physical states.
– A physical state has multiple descriptions.
– Transformation changes descriptions.
• Global Symmetry
– Transformation changes physical states.
This is a more fundamental distinction than spacetime-dependence.
Non-Abelian Gauge Symmetry
Modification/Generalization?
d A
(1)= dL
(0)+[A
(1), L
(0)] F
(2)= dA
(1)+ A
(1)A
(1)Þ d F
(2)= [F
(2), L
(0)]
Non-Commutative Gauge Theory
• D-brane world-volume theory in B-field background. [Chu-Ho 99, Seiberg-Witten 99]
• Commutation relation determined by B-field.
• U(1) gauge theory is non-Abelian.
[xi, xj] = iQij
Lie 3-Algebra Gauge Symmetry
• BLG model for multiple M2-branes [07].
• needed for manifest compatibility with SUSY.
(ABJM model does not have manifest full SUSY.)
[Ta, Tb, Tc] = f abcdTd A = Aiab(Ta Ä Tb)dxi
(Dif)a = ¶ifa + Aibcfd f bcda
Generalization of Lie 3-algebra
• All Lie n-algebra gauge symmetries are special cases of ordinary (Lie algebra) non-Abelian
gauge symmetries.
[Ta1 , Ta2 ,, Tan] = f a1a2anan+1Tan+1 A = Aa1a2an-1 (Ta1 Ä Ta2 ÄÄ Tan-1 ) f = faTa
[A, f] = Aa1a2an-1fa
n f a1a2anan+1Tan+1
More Gauge Theories
• Abelian Higher-form gauge theories
• Self-dual gauge theories
• Non-Abelian higher-form gauge theories
• NA SD HF GT w. Lie 3 (on NC space?)
A(n) = n!1 Ai1i2indxi1 Ùdxi2 ÙÙdxin F(n+1) = dA(n)
dA(n) = dL(n-1) dF(n+1) = 0
Dirac Monopole
• Jacobi identity (associativity)
is violated in the presence of magnetic monopoles.
• Distribution of magnetic monopoles gauge bundle does not exist
2-form gauge potential (3-form field strength) (Abelian bundle Abelian gerbe)
Q: How to generalize to non-Abelian gauge theory?
[[Di, Dj ], Dk] +[[Dj, Dk], Di] +[[Dk, Di], Dj] = 0
Self-Dual Gauge Fields
• Examples:
type IIB supergravity type IIB superstring M5-brane theory twistor theory
• D=4k+2 (4k) for Minkowski (Euclidean) spacetime.
Self-Dual Gauge Fields
• When D = 2n, the self-duality condition is
• a.k.a. chiral gauge bosons.
• How to produce 1st order diff. eq. from action?
Trick: introduce additional gauge symmetry
F = *F
F = n!1 Fi1i2in dxi1 Ù dxi2 ÙÙ dxin
*F = n!1 (D-n)!1 e i1i2iD F i1i2in dxin+1 Ù dxin+2 ÙÙ dxiD
Chiral Boson in 2D
[Floreanini-Jackiw ’87]
• Self-duality
• Lagrangian
• Gauge symmetry
• Euler-Lagrange eq.
• Gauge transformation
• 1-1 map btw space of sol’s and chiral config’s
f ˙ - ¢ f = 0
L = 12 f ¢ ( ˙ f - ¢ f )
¶x(¶t -¶x)f = 0
Þ (¶t -¶x)f = g(t)
• Due to gauge symmetry, f is not an observable.
• k=f’ is an observable the Lagrangian is
• Formal nonlocality is unavoidable, although physics is local.
• Equivalent to a Weyl spinor in 2D in terms of the density of solitons.
• Lorentz symmetry is hidden.
L = 12 k(t, x)
( ò dye(x- y) ˙ k (t, y) - k(t, x))
[
f
(t, x),f
(t, y)] = ie
(x- y)Comments
• No vector potential needed for gauge symmetry Vector potential is useful for defining
cov. quantities from deriv. of cov. quantities.
But it is not absolutely necessary.
• The formulation can be generalized to self-dual higher-form gauge theories.
Dm = ¶m + Am
Example: M5-brane theory (D=5+1)
[Howe-Sezgin 97, Pasti-Sorokin-Tonin 97, Aganagic-Part- Popescu-Schwarz 97, …]
[Chen-Ho 10]
[Ho-Imamura-Matsuo 08]
Questions
• What is the geometry of Abelian higher-form gauge symmetry?
(bundles gerbes?)
• How to define non-Abelian higher-form gauge symmetry?
• Can we generalize the notion of covariant derivative and field strength?
• Do we still need the covariant derivative?
Non-Abelian Self-Dual 2-Form Gauge Potential
• M5-brane in the (3-form) C-field background.
[Ho-Matsuo 08, Ho-Imamura-Matsuo-Shiba 08]
• Non-Abelian gauge symmetry for 2-form gauge potential
• Nambu-Poisson algebra (ex. of Lie 3-algebra) (Volume-Preserving Diffeomorphism)
• Part of the gauge potential VPD
• 1 gauge potential for 2 gauge symmetries!
[Ho-Yeh 11]
Non-Local Non-Abelian
Self-Dual 2-Form Gauge Field
• Need non-locality to circumvent no-go thms for multiple M5-branes. [Ho-Huang-Matsuo 11]
• In “5+1” formulation, decompose all fields
into zero modes vs. non-zero modes in the 5-th dimension.
• Zero modes 1-form potential A in 5D
• Non-zero modes 2-form potential B in 5D
Comments
• Self-duality: Instantons, Penrose’s twistor theory applied to Maxwell and GR.
• In 4D, 2-form ≈ 0-form, 3-form ≈ (-1)-form, 4-form ≈ trivial through EM duality.
• Expect more new gauge symmetries from string theory to be discovered.
• “Symmetry dictates interaction” -- C. N. Yang.